Difference Methods to One and Multidimensional Interdiffusion Models With Vegard Rule

Abstract

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.

Compendium About Multicomponent Interdiffusion in Two Dimensions

“…; s: ½ Now we formulate 2-D and 3-D models. [24][25][26] Let X Ì R n , n = 2, 3. In these cases, the volume continuity equation (Eq.…”

“…[16], [9], and [17]). [25] B. Difference Scheme in Two Dimensions To find numerical solutions for the model, we use the implicit finite difference method (FDM).…”

“…The modification with the use of Eq. [25] allowed construction of the Vegard rule preserving the implicit finite difference method, generated by some linearization and splitting ideas used, to present the numerical solutions. Numerical methods that preserve some futures of a continuous model are very important and interesting from a physical and mathematical point of view.…”

“…For details concerning the existence and uniqueness of the discrete solutions and other properties of the numerical method, we refer the readers to our previous articles. [25,26] C. Data Given the model and numerical method, the computations were made to simulate the interdiffusion provided in the experiment. The necessary data are given in Table I.…”

“…Taking benefit of the concept of the drift potential, we formulated the nonlinear parabolic-elliptic system of strongly coupled differential equations and constructed the implicit difference methods to simulate the interdiffusion in a multicomponent system. [24][25][26] Exemplary numerical calculations were made for ternary systems of 2-D geometry. In Reference 25, the component diffusivities were assumed composition independent.…”